The Ilieff-Sendov Conjecture Robert "Dr. Bob" Gardner Fall 2011
The Ilieff-Sendov Conjecture Robert "Dr. Bob" Gardner Fall 2011
The Ilieff-Sendov Conjecture Robert "Dr. Bob" Gardner Fall 2011
We can represent a closed half plane with the equation Im((z − a)/b) ≤ 0.
This represents the half plane to the right of the line Im((z − a)/b) = 0 when
traveling along the line in the “direction” of b. This representation, along
with some standard properties of logarithms and derivatives in the complex
setting, allow us to prove the following so-called Gauss-Lucas Theorem (or
sometimes simply the Lucas Theorem). For a reference, see page 29 of [1].
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So
Suppose the half plane H that contains all the zeros of P (z) is described by
Im((z − a)/b) ≤ 0. Then Im((rk − a)/b) ≤ 0 for k = 1, 2, . . . , n. Now let z ∗
be some number not in H. We want to show that P 0(z ∗ ) 6= 0 (this will mean
that all the zeros of P 0(z) are in H). Well, Im((z ∗ − a)/b) > 0. Let rk be
some zero of P . Then
∗ ∗
z − rk z − a − rk + a
Im = Im
b b
∗
z −a rk − a
= Im − Im > 0.
b b
(Notice that Im((z ∗ − a)/b) > 0 since z ∗ is not in H, and −Im((rk − a)/b) ≥ 0
since rk is in H.) The imaginary parts of reciprocal numbers have opposite
signs, so Im(b/(z ∗ − rk )) < 0. Hence, by applying (∗),
0 ∗ X n
bP (z ) b
Im = Im ∗ < 0.
P (z ∗) z − rk
k=1
P 0 (z ∗)
So ∗
6= 0 and P 0(z ∗ ) 6= 0. Therefore, if P 0(z) = 0, then z ∈ H.
P (z )
Corollary 1. The convex polygon in the complex plane which contains all
the zeros of a polynomial P , also contains all the zeros of P 0.
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Note. For example, if P has eight zeros, then the convex polygon containing
them might look like the following.
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Corollary 3. If all the zeros of a polynomial P lie in |z| ≤ 1, then all the
zeros of P 0 also lie in |z| ≤ 1.
Note. It is important that we are studying the set of all polynomials with
their zeros in |z| ≤ 1. We can violate Corollary 3 by considering a non-
polynomial. Consider, for example, f (z) = zez/2 . Then the only zero of f is
z = 0, and so all the zeros of f lie in |z| ≤ 1. However, f 0 (z) = ez/2 + 12 zez/2 =
( 12 z + 1)ez/2 . Then f 0 has a zero at z = −2 and so there is a zero outside of
|z| ≤ 1.
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−an (r1 + r2 + · · · + rn ) and the centroid of the zeros of P is
r1 + r2 + · · · + rn 1 −an−1 −an−1
= = .
n n an nan
Let the zeros of P 0 be s1 , s2 , . . . , sn−1 . Then
n
X n−1
Y
0 k−1
P (z) = kak z = nan (z − sk ).
k=1 k=1
Note. Now for the object of our interest! It is known variously as the Ilieff
Conjecture, the Ilieff-Sendov Conjecture, and the Sendov Conjecture (making
it particularly difficult to search for papers on the subject). It was originally
posed by Bulgarian mathematician Blagovest Sendov in 1958 (according to
[12]; sometimes the year 1962 is reported [11]), but often attributed (as Miller
says in [11]) to Ilieff because of a reference in Hayman’s Research Problems
in Function Theory in 1967 [8].
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Note. Combining the Ilieff-Sendov Conjecture with Corollary 3, we can
further restrict the conjectured location of the critical points of P .
Note. According to a 2008 paper by Michael Miller [12], there have been over
80 papers written on the conjecture. As a result, it has been demonstrated
in many special cases. Some of the special cases are (where we understand
that all polynomials have their zeros in |z| ≤ 1):
4. If the convex hull containing the roots of a polynomial p have its vertices
on |z| = 1 then p satisfies the conjecture [15],
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5. polynomials with real and non-positive coefficients [16],
Note. Many of the papers on the conjecture have been a bit more computa-
tional and have centered around finding “extremal” polynomials which push
the locations of the zero of P to the edge of the |z − r| ≤ 1 region. This is
especially true of the recent work of Micheal Miller [11, 12].
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However, this conjecture is not true as shown by Micheal Miller in 1990
[11]. The following eighth degree polynomial violates the Goodman, Rahman,
Ratti Conjecture:
Miller also found degree 6, 10, and 12 polynomials violating the new conjec-
ture.
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approach. . . maybe with good reason!
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Bibliography
[1] Ahlfors, L., Complex Analysis: An Introduction to the Theory of Analytic Functions
of One Complex Variable, Third Edition, McGraw-Hill: 1979.
[2] Bojanov, B., Rahman, Q., and J. Szynal, “On a Conjecture of Sendov about the
Critical Points of a Polynomial,” Mathematische Zeitschrift, 190, 281–285 (1985).
[3] Borcea, I., “On the Sendov Conjecture for Polynomials with at Most Six Distinct
Roots,” Journal of Mathematical Analysis and Applications, 200, 182–206 (1996).
[4] Brown, J., “On the Sendov Conjecture for Sixth Degree Polynomials,” Proceedings of
the American mathematical Society, 113(4), 939–946 (1991).
[5] Brown, J. and G. Xiang, “Proof of the Sendov Conjecture for Polynomials of Degree at
Most Eight, Journal of Mathematical Analysis and Applications, 232, 272–292 (1999).
[6] Cohen, G. and G. Smith, “A Simple Verification of Ilieff’s Conjecture for Polynomials
With Three Zeros,” The American Mathematical Monthly, 95(8), 734–737 (1988).
[7] Goodman, A., Q. Rahman, and J. Ratti, “On the Zeros of a Polynomial and its
Derivative,” Proceedings of the American Mathematical Society, 21, 273–274 (1969).
[8] Hayman, W., Research Problems in Function Theory, London: Athlone (1967).
[9] Marden, M., Geometry of Polynomials, Mathematical Monographs and Surveys #3,
AMS: 1986.
[10] Meir, A. and A. Sharma, “On Ilyeff’s Conjecture,” Pacific Journal of Mathematics,
31, 459–467, (1969).
[11] Miller, M., “Maximal Polynomials and the Illieff-Sendov Conjecture, Transactions of
the American Mathematical Society, 321(1), 285–303 (1990).
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[12] Miller, M., “Unexpected Local Extrema for the Sendov Conjecture,” Journal of Math-
ematical Analysis and Applications, 348, 461–468 (2008).
[13] Rubenstein, Z., “On a Problem of Ilyeff,” Pacific Journal of Mathematics, 26, 159–161
(1968).
[14] Saff, E. and J. Twomey, “A Note on the Location of Critical Points of Polynomials,”
Proceedings of the American Mathematical Society, 27, 303-308 (1971)
[16] Schmeisser, G., “Zur Lage der Kritichen punkte eines polynoms,” Rend. Sem. Mat.
Univ. Padova, 46, 405–415 (1971).
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